economics 404

a) Using the equilibrium conditions above, show that this model can be represented by the standard 3 equations (13) #1 = BE (#1+1) + Kyl (14) (15) plus expressions for the natural rate of output: (16) the natural real rate of interest: F - -o(1- P) (17) and the output gap ye = yo - " (18) where a, follows the process in equation 12. (Hint: You may want to start by finding the natural rate of output (equation 16) and then writing the household Euler equation in terms of (linearized) output, the natural rate of output, the nominal interest rate, TFP and expected inflation.) b) Using the method of undetermined coefficients, find the response of the output gap and inflation to an exogenous increase in a, when prices are sticky and monetary policy follows the Taylor Rule above. To do this, guess that the solution for each variable is a linear function of the shock at: ye = Ayat c) Interpret your results in (b). In particular, carefully explain how, and why, TFP shocks affect the output gap and inflation in this model. d) Instead of following the Taylor Rule above, policy is now set optimally. One result is that, under optimal policy, the real interest rate tracks the natural real interest rate. From your knowledge of this model and optimal policy, what is the optimal path for the output gap and inflation in response to a TFP shock and why (you do not need to derive anything)? e) Suppose the monetary policymaker wants to implement optimal policy using an interest rate rule for i. Explain why the policymaker cannot simply use a rule which attempts to set the nominal interest rate (i) equal to the natural real interest rate ("). What other component(s) should the rule contain and why (you do not need to derive anything)?Question 4 (20 points) This question considers a TFP shock in the New Keynesian model. The representative household's utility function is: Nite It * (4) In linearized form, the equilibrium conditions for this model are as follows. The house- hold's Euler equation and labor supply conditions are: (5) (6) The linearized equilibrium conditions for firms are: ye = at + nut mc = wt - at (8) it = BE,(it+1) + Amic (9) The resource constraint is: yt = C (10) Monetary policy follows a simple Taylor Rule: (11) (Linearized) TFP follows an AR(1) process (12) et is i.i.d. In percentage deviations from steady state: me, is real marginal cost, & is con- sumption, w, is the real wage, n, is hours worked, y, is output and a, is Total Factor Productivity. In deviations from steady state: i, is the nominal interest rate, #, is inflation. A is a function of model parameters, including the degree of price stickiness. Assume that o, > 1, 0 > 0) always exist? If not, describe the set of parameter values (including the policy parameter i) for which such an equilibrium exists. Finally, define the welfare function of this economy as the measure of the various KW market meetings times the net surplus generated in each meeting, i.e., W = olu(go) - qo] + (1 - o)[u(q1) - qi]. g) Can you describe the sign of the term OW/do for the various values of g74Question 2 (20 points) This question studies the co-existence of money and credit. Time is discrete with an infinite horizon. Each period consists of two subperiods. In the day, trade is partially bilateral and anonymous as in Kiyotaki and Wright (1991) (call this the KW market). At night trade takes place in a Walrasian or centralized market (call this the CM). There are two types of agents, buyers and sellers, and the measure of both is normalized to 1. The per period utility for buyers is u(q) + U(X) - H, and for sellers it is -q + U(X) - H, where q is the quantity of the day good produced by the seller and consumed by the buyer, X is consumption of the night good (the numeraire), and # is hours worked in the CM. In the CM, all agents have access to a technology that turns one unit of work into a unit of good. The functions u, U satisfy the usual assumptions; I will only spell out the most crucial ones: There exists X* E (0, co) such that U'(X*) = 1, and we define the first-best quantity traded in the KW market as q' = (q : u'(q') = 1}. The difference compared to the baseline model is that there are two types of sellers. Type-0 sellers, with measure o E [0, 1], accept credit. More precisely, in meetings with a type-0 seller (type-0 meetings), no medium of exchange (MOE) is necessary, and the buyer can purchase day good by promising to repay the seller in the forthcoming CM with numeraire good (this arrangement is called an IOU). The buyer can promise to repay any amount (no credit limit), and her promise is credible (buyers never default). Type-1 sellers, with measure 1 - o, never accept credit, hence, any purchase of the day good must be paid for on the spot (quid pro quo) with money. All buyers meet a seller in the KW market, so that o is the probability with which a buyer meets a type-0 seller, and 1 - o is the probability with which she meets a type-1 seller. The rest is standard. Goods are non storable. There exits a storable and rec- ognizable object, fiat money, that can serve as a MOE in type-1 meetings. Money supply is controlled by a monetary authority, and we consider simple policies of the form Miti = (1 + #)M, u > 8 - 1. New money is introduced, or withdrawn if # 0 per unit of time, and while a firm is searching for a worker it has to pay a search (or recruiting) cost, pc > 0, per unit of time. Firms that are training their workers do not pay this cost (they are done recruiting). Productive jobs are exogenously destroyed at rate > > 0 (only productive jobs are subject to this shock; matches at the training stage cannot be terminated). All agents discount future at the rate r > 0, and unemployed workers enjoy a benefit > > 0 per unit of time. While at the training stage the worker does not receive an unemployment benefit (a trainee is not unemployed). a) Define the value function of the typical firm for all the possible states of the world it may find itself in. b) Do the same for the typical worker. c) Combine the free entry condition with the expressions you provided in part (a) in order to derive the job creation (JC) curve of this economy. d) Using the same methodology as in the lectures (adjusted to accommodate the differences in the new environment), derive the wage curve (WC) for this economy. 1 Hence, two parties who met at time, say, f are negotiating over an object that will be paid in the future (at time f + 1/a, in expected terms). But, as is always the case, the Nash Bargaining problem is to split the generated surplus as of time f